ACT Math: Reviewing Law of Cosines & Law of Sines

Many students prefer the Math section of the ACT to that of the SAT because they view it as being more straightforward with questions that more closely resemble what they might see on a test in school.  This, however, is not the only difference between the two tests; the ACT asks questions that go deeper into the curriculum.  You are likely to encounter questions that come from both Algebra 2 and Pre-Calculus/Trigonometry on the ACT whereas the SAT contains content almost exclusively from Pre-Algebra, Algebra 1 and Geometry.

If you want to achieve a high score on the ACT Math (anything in the 30s), you should familiarize yourself with the topics from Algebra 2 and Trigonometry that are commonly tested.  Among these are both the Law of Cosines and the Law of Sines.  Let’s look at each of these.

The Law of Cosines defines a relationship between the sides of any triangle and its angles. The formula is:

$c^{2}=a^{2}+b^{2}-2ab\cdot cosC$

If you’re not familiar with this formula (and even if you are), it’s important to know that the lower-case letters represent sides and the upper-case letters represent angles.  You also need to know that each side is opposite its corresponding angle.  For example, side c is opposite angle C.

The Law of Cosines can be used for two things:
1)  Finding a missing side length provided you know the lengths of the two other sides and one of the angles;
2)  Finding any of the angles if you know the lengths of all three sides.

Consider the following problem involving the Law of Cosines (and, yes, they really do give you the formula on the test!):

You want to find the distance across a pond from Point A to Point B as shown in the diagram below.  You measure from a point on land and determine that the distance to Point A is 237 feet and the distance to Point B is 185 feet.  The angle between the straight lines from this point to Points A and B is 126 degrees.  What is the approximate distance across the lake?

(Note:  The law of cosines states that for any triangle with vertices AB, and C and the sides opposite those vertices with lengths ab, and c, respectively, c2 = a2 + b2 – 2ab cos C .)

(A)  301 feet
(B)  327 feet
(C)  355 feet
(D)  377 feet
(E)  422 feet

Solving this problem involves plugging numbers into the proper places in the formula and then doing the arithmetic.

c2 = a2 + b2 – 2ab cos C
c2 = 2372 + 1852 – 2(237)(185) cos(126)
c2 = 141936.89
c = 376.7

The correct answer is Choice (D).

A second formula, the Law of Sines, gives a different relationship between the sides and angles.  This formula states

$\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$

The Law of Sines can be used to:
1)  Find the measure of an angle when you know the lengths of two sides and the measure of an angle:
2)  Find the measure of a side opposite a known angle when you know the length of one side and the measure of two angles.

Consider the following problem that involves the Law of Sines.

Barbara is trying to determine the proper flight path for an upcoming trip.  Her map shows the distances between Allen, Brickton and Centerville as shown in the diagram below. She knows from a previous trip that the angle formed by the straight line distances from Allen to Brickton and Brickton to Centerville is 63 degrees.  What is the approximate angle formed by the straight line distances from Allen to Brickton and Allen to Centerville, as marked by the x in the diagram?

(Note: The law of sines states for any triangle with vertices AB, and C and the sides opposite those vertices with lengths ab, and c, respectively, $\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$ )

(A)  65 degrees
(B)  71 degrees
(C)  77 degrees
(D)  79 degrees
(E)  85 degrees

Once again, you can solve this problem by plugging numbers into the proper places in the formula (you only need two of the fractions) and then doing the arithmetic.

$\frac{sinA}{a}=\frac{sinB}{b}$
$\frac{sin63}{512}=\frac{sinx}{560}$
$512 \, sinx=560\, sin63$
$sin x =\frac{560sin63}{512}$
$sin x = .974538$
$x=$ Sin-1 .974538
$x = 77.04$

The correct answer is Choice (C).